A000359 -id:A000359 - cp's OEIS frontend

A003713 Expansion of e.g.f. log(1/(1+log(1-x))).

0, 1, 2, 7, 35, 228, 1834, 17582, 195866, 2487832, 35499576, 562356672, 9794156448, 186025364016, 3826961710272, 84775065603888, 2011929826983504, 50929108873336320, 1369732445916318336, 39005083331889816960, 1172419218038422659456, 37095226237402478348544

Offset: 0

N. J. A. Sloane, R. H. Hardin, Simon Plouffe

a(n+1) is the permanent of the n X n matrix M with M(i,i) = i+1, other entries 1. - Philippe Deléham, Nov 03 2005
Supernecklaces of type III (cycles of cycles). - Ricardo Bittencourt, May 05 2013
Unsigned coefficients for the raising / creation operator R for the Appell sequence of polynomials A238385: R = x + 1 - 2 D + 7 D^2/2! - 35 D^3/3! + ... . - Tom Copeland, May 09 2016

J. Ginsburg, Iterated exponentials, Scripta Math., 11 (1945), 340-353.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n = 0..100
P. J. Cameron, Sequences realized by oligomorphic permutation groups, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.
Philippe Flajolet and Robert Sedgewick, Analytic Combinatorics, Cambridge Univ. Press, 2009, page 125.
Jekuthiel Ginsburg, Iterated exponentials, Scripta Math., 11 (1945), 340-353. [Annotated scanned copy]
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 34
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 298

a(n)=|A039814(n, 1)| (first column of triangle). Cf. A000268, A000310, A000359, A000406, A001765.
Cf. A007840, A089064.
Cf. A238385.

series(ln(1/(1+ln(1-x))),x,17);
with (combstruct): M[ 1798 ] := [ A,{A=Cycle(Cycle(Z))},labeled ]:

With[{nn=20},CoefficientList[Series[Log[1/(1+Log[1-x])],{x,0,nn}],x]Range[0,nn]!] (* Harvey P. Dale, Dec 15 2012 *)
Table[Sum[(-1)^(n-k) * (k-1)! * StirlingS1[n, k], {k, 1, n}], {n, 0, 20}] (* Vaclav Kotesovec, Apr 19 2024 *)

a(n)=if(n<0,0,n!*polcoeff(-log(1+log(1-x+x*O(x^n))),n))

Sum_{k=1..n} (k-1)!*|Stirling1(n, k)|. - Vladeta Jovovic, Sep 14 2003
a(n+1) = n! * Sum_{k=0..n} A007840(k)/k!. E.g., a(4) = 228 = 24*(1/1 + 1/1 + 3/2 + 14/6 + 88/24) = 24 + 24 + 36 + 56 + 88. - Philippe Deléham, Dec 10 2003
a(n) ~ (n-1)! * (exp(1)/(exp(1)-1))^n. - Vaclav Kotesovec, Jun 21 2013
a(0) = 0; a(n) = (n-1)! + Sum_{k=1..n-1} binomial(n-1,k) * (k-1)! * a(n-k). - Ilya Gutkovskiy, Jul 18 2020

Thanks to Paul Zimmermann for comments.

A000268 E.g.f.: -log(1+log(1+log(1-x))).

Original entry on oeis.org

1, 3, 15, 105, 947, 10472, 137337, 2085605, 36017472, 697407850, 14969626900, 352877606716, 9064191508018, 252024567201300, 7542036496650006, 241721880399970938, 8261159383595659128, 299916384730043070880, 11526945327529620432872, 467583770376898192016104

Offset: 1

N. J. A. Sloane

nonn

J. Ginsburg, Iterated exponentials, Scripta Math., 11 (1945), 340-353.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Seiichi Manyama, Table of n, a(n) for n = 1..400 (terms 1..100 from T. D. Noe)
P. J. Cameron, Sequences realized by oligomorphic permutation groups, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.
Jekuthiel Ginsburg, Iterated exponentials, Scripta Math., 11 (1945), 340-353. [Annotated scanned copy]
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 299
Kruchinin Vladimir Victorovich, Composition of ordinary generating functions, arXiv:1009.2565 [math.CO], 2010.

a(n)=|A039815(n, 1)| (first column of triangle).

Cf. A003713, A000310, A000359, A000406, A001765.

max = 20; CoefficientList[-Log[1 + Log[1 + Log[1 - x]]]/x + O[x]^max, x] * Range[max]! (* Jean-François Alcover, Feb 06 2016 *)

a(n) = sum(m=1, n, (m-1)!*(-1)^(n+m)*sum(k=m, n, stirling(n,k,1)*stirling(k,m,1))); \\ Michel Marcus, Feb 06 2016

a(n) = sum((m-1)!*(-1)^(n+m)*sum(stirling1(n, k)*stirling1(k, m), k,m,n), m,1,n), n>0. - Vladimir Kruchinin, Sep 14 2010

Revised description from Christian G. Bower, Aug 15 1998

A000310 Coefficients of iterated exponentials.

Original entry on oeis.org

1, 4, 26, 234, 2696, 37919, 630521, 12111114, 264051201, 6445170229, 174183891471, 5164718385337, 166737090160871, 5822980248613990, 218756388226681557, 8797723991458469015, 377159237609540937788, 17170729962232112834302, 827382365085791968518198, 42070004707327023844695198

Offset: 1

N. J. A. Sloane

nonn

J. Ginsburg, Iterated exponentials, Scripta Math., 11 (1945), 340-353.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n=1..100
P. J. Cameron, Sequences realized by oligomorphic permutation groups, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.
Jekuthiel Ginsburg, Iterated exponentials, Scripta Math., 11 (1945), 340-353. [Annotated scanned copy]
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 300

a(n) = |A039816(n, 1)| (first column of triangle). Cf. A003713, A000268, A000359, A000406, A001765.

max = 20; CoefficientList[-Log[1 + Log[1 + Log[1 + Log[1 - x]]]]/x + O[x]^max, x]*Range[max]! (* Jean-François Alcover, Feb 07 2016 *)

T(n, k) = if(k==1, (n-1)!, sum(j=1, n, abs(stirling(n, j, 1))*T(j, k-1)));
a(n) = T(n, 4); \\ Seiichi Manyama, Feb 11 2022

my(x='x+O('x^40)); Vec(serlaplace(-log(1+log(1+log(1+log(1-x)))))); \\ Michel Marcus, Feb 11 2022

E.g.f.: -log(1+log(1+log(1+log(1-x)))).

A000406 Coefficients of iterated exponentials.

Original entry on oeis.org

1, 6, 57, 741, 12244, 245755, 5809875, 158198200, 4877852505, 168055077875, 6400217406500, 267058149580823, 12118701719205803, 594291742526530761, 31323687504696772151, 1766116437541895988303, 106080070002238888908150

Offset: 1

N. J. A. Sloane

nonn

J. Ginsburg, Iterated exponentials, Scripta Math., 11 (1945), 340-353.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n=1..100
P. J. Cameron, Sequences realized by oligomorphic permutation groups, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.
Jekuthiel Ginsburg, Iterated exponentials, Scripta Math., 11 (1945), 340-353. [Annotated scanned copy]
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 302

Cf. A003713, A000268, A000310, A000359, A001765.

With[{nn=20},CoefficientList[Series[-Log[1+Log[1+Log[1+Log[1+ Log[1+ Log[1- x]]]]]],{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Nov 03 2015 *)

T(n, k) = if(k==1, (n-1)!, sum(j=1, n, abs(stirling(n, j, 1))*T(j, k-1)));
a(n) = T(n, 6); \\ Seiichi Manyama, Feb 11 2022

my(N=20, x='x+O('x^N)); Vec(serlaplace(-log(1+log(1+log(1+log(1+log(1+log(1-x)))))))) \\ Seiichi Manyama, Feb 11 2022

E.g.f.: -log(1+log(1+log(1+log(1+log(1+log(1-x)))))).

A001765 Coefficients of iterated exponentials.

Original entry on oeis.org

1, 7, 77, 1155, 21973, 506989, 13761937, 429853851, 15192078027, 599551077881, 26140497946017, 1248134313062231, 64783855286002573, 3632510833677434324, 218845138322691595694, 14099918095287618382033, 967508237903439910445565, 70447525748137979196484589

Offset: 1

N. J. A. Sloane

J. Ginsburg, Iterated exponentials, Scripta Math., 11 (1945), 340-353.
T. Hogg and B. A. Huberman, Attractors on finite sets: the dissipative dynamics of computing structures, Phys. Review A 32 (1985), 2338-2346.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n=1..100
P. J. Cameron, Sequences realized by oligomorphic permutation groups, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.
T. Hogg and B. A. Huberman, Attractors on finite sets: the dissipative dynamics of computing structures, Phys. Review A 32 (1985), 2338-2346. (Annotated scanned copy)
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 303

Cf. A003713, A000268, A000310, A000359, A000406.

With[{nn=20},CoefficientList[Series[-Log[1+Log[1+Log[1+Log[1+Log[1+Log[1+Log[1-x]]]]]]],{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Jan 07 2023 *)

T(n, k) = if(k==1, (n-1)!, sum(j=1, n, abs(stirling(n, j, 1))*T(j, k-1)));
a(n) = T(n, 7); \\ Seiichi Manyama, Feb 11 2022

my(N=20, x='x+O('x^N)); Vec(serlaplace(-log(1+log(1+log(1+log(1+log(1+log(1+log(1-x))))))))) \\ Seiichi Manyama, Feb 11 2022

E.g.f.: -log(1+log(1+log(1+log(1+log(1+log(1+log(1-x))))))).

A039817 Triangle read by rows: matrix 5th power of the Stirling-1 triangle A008275.

Original entry on oeis.org

1, -5, 1, 40, -15, 1, -440, 235, -30, 1, 6170, -4200, 775, -50, 1, -105315, 86020, -20475, 1925, -75, 1, 2120610, -2001055, 577570, -70525, 4025, -105, 1, -49242470, 52305780, -17609620, 2623145, -195300, 7490, -140, 1, 1296133195, -1520815230, 581516560, -101595060, 9264045, -464940, 12810, -180, 1

Offset: 1

Christian G. Bower, Feb 15 1999

			Triangle begins:
        1;
       -5,     1;
       40,   -15,      1;
     -440,   235,    -30,    1;
     6170, -4200,    775,  -50,   1;
  -105315, 86020, -20475, 1925, -75, 1;
  ...

Seiichi Manyama, Rows n = 1..140, flattened

Cf. A000359 (first column), A008275.

Cf. A039813, A039814, A039815, A039816.

Flatten[Table[SeriesCoefficient[(Log[1+Log[1+Log[1+Log[1+Log[1+x]]]]])^k,{x,0,n}] n!/k!, {n,9}, {k,n}]] (* Stefano Spezia, Sep 12 2022 *)

E.g.f. of k-th column: ((log(1+log(1+log(1+log(1+log(1+x))))))^k)/k!.

A302358 a(n) = coefficient of x^n in the n-th iteration (n-fold self-composition) of e.g.f. -log(1 - x).

Original entry on oeis.org

1, 2, 15, 234, 6170, 245755, 13761937, 1030431500, 99399019626, 12003835242090, 1773907219147800, 314880916127332489, 66109411013740671200, 16204039283106534720952, 4585484528618722750937783, 1483746673734716952089913364, 544359300175753347889146067840

Offset: 1

Ilya Gutkovskiy, Apr 06 2018

nonn

			The initial coefficients of successive iterations of e.g.f. A(x) = -log(1 - x) are as follows:
n = 1: 0, (1), 1,   2,    6,    24,  ... e.g.f. A(x)
n = 2: 0,  1, (2),  7,   35,   228,  ... e.g.f. A(A(x))
n = 3: 0,  1,  3, (15), 105,   947,  ... e.g.f. A(A(A(x)))
n = 4: 0,  1,  4,  26, (234), 2696,  ... e.g.f. A(A(A(A(x))))
n = 5: 0,  1,  5,  40,  440, (6170), ... e.g.f. A(A(A(A(A(x)))))

Seiichi Manyama, Table of n, a(n) for n = 1..246
Jekuthiel Ginsburg, Iterated exponentials, Scripta Math., 11 (1945), 340-353. [Annotated scanned copy]
N. J. A. Sloane, Transforms

Cf. A000268, A000310, A000359, A000406, A001765, A003713, A104150, A139383, A158832, A174482, A261280.

g:= x-> -log(1-x):
a:= n-> n! * coeff(series((g@@n)(x), x, n+1), x, n):
seq(a(n), n=1..19);  # Alois P. Heinz, Feb 11 2022

Table[n! SeriesCoefficient[Nest[Function[x, -Log[1 - x]], x, n], {x, 0, n}], {n, 17}]

T(n, k) = if(k==1, (n-1)!, sum(j=1, n, abs(stirling(n, j, 1))*T(j, k-1)));
a(n) = T(n, n); \\ Seiichi Manyama, Feb 11 2022

a(n) = T(n,n), T(n,k) = Sum_{j=1..n} |Stirling1(n,j)| * T(j,k-1), k>1, T(n,1) = (n-1)!. - Seiichi Manyama, Feb 11 2022

A111933 Triangle read by rows, generated from Stirling cycle numbers.

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 1, 3, 7, 6, 1, 4, 15, 35, 24, 1, 5, 26, 105, 228, 120, 1, 6, 40, 234, 947, 1834, 720, 1, 7, 57, 440, 2696, 10472, 17582, 5040, 1, 8, 77, 741, 6170, 37919, 137337, 195866, 40320, 1, 9, 100, 1155, 12244, 105315, 630521, 2085605, 2487832, 362880

Offset: 1

Gary W. Adamson, Aug 21 2005

Let M = the infinite lower triangular matrix of Stirling cycle numbers (A008275). Perform M^n * [1, 0, 0, 0, ...] forming an array. Antidiagonals of that array become the rows of this triangle.

			Row 5 of the triangle = 1, 4, 15, 35, 24; generated from M^n * [1,0,0,0,...] (n = 1 through 5); then take antidiagonals.
Terms in the array, first few rows are:
  1, 1,  2,   6,    24,    120, ...
  1, 2,  7,  35,   228,   1834, ...
  1, 3, 15, 105,   947,  10472, ...
  1, 4, 26, 234,  2697,  37919, ...
  1, 5, 40, 440,  6170, 105315, ...
  1, 6, 57, 741, 12244, 245755, ...
  ...
First few rows of the triangle are:
  1;
  1, 1;
  1, 2,  2;
  1, 3,  7,   6;
  1, 4, 15,  35,  24;
  1, 5, 26, 105, 228,  120;
  1, 6, 40, 234, 947, 1834, 720;
  ...

Seiichi Manyama, Antidiagonals n = 1..140, flattened

Row 1..7 give A000142(n-1), A003713, A000268, A000310, A000359, A000406, A001765.
Column 3 of the array = A005449.
Column 4 of the array = A094952.
Cf. A008275, A302358.

a(28), a(36) and a(45) corrected by Seiichi Manyama, Feb 11 2022

A292691 a(n) = C(A001359(n)), n >= 1, with C(n) = (4((n-1)! + 1) + n)/(n(n+2)) for n >= 2.

Original entry on oeis.org

1, 3, 101505, 259105757127, 1356566605613854774200240267, 1851197466245939272480116323530608949000567215

Offset: 1

Jaime Gómez, Sep 20 2017

nonn

Clement's criterion for twin primes is, for integers with n >= 2: n and n + 2 are both primes if and only if 4*((n-1)! + 1) + n == 0 (mod n*(n+2)). See the Clement and Ribenboim links. Like the criteron for primality using Theorem 81 of Hardy and Wright, p. 69, it "is of course quite useless as a practical test".
a(n) is an integer because of the necessary part of this twin prime criterion.
Thanks to Wolfdieter Lang for comments and helpful advice.

			a(2) = 3, because A001359(2) = 5 and C(5) = (4*(4! + 1) + 5)/(5*7) = 3.
a(2) = 3 because A014574(2) = 6 and delta(6) = (4*4! + 6 + 3)/35 = 3.

G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, Oxford Science Publications, 1979.
P. Ribenboim, The New Book of Prime Number Records, Springer-Verlag NY 1996, pp. 259-260 (a proof of Clement's theorem).

Jaime Gómez, Table of n, a(n) for n = 1..23
P. A. Clement, Congruences for sets of primes, American Mathematical Monthly, vol. 56 (1949), pages 23-25.
L. Cong and Z. Li, On Wilson's Theorem and Polignac's Conjecture, arXiv:math/0408018 [math.NT], 2004.

Cf. A001359, A007619, A014574.

p1[1] = 3; p1[n_] := p1[n] = (p = NextPrime[p1[n-1]]; While[!PrimeQ[p + 2], p = NextPrime[p]]; p);
a[n_] := (4*((p1[n] - 1)! + 1) + p1[n])/(p1[n]*(p1[n] + 2));
Array[a, 6] (* Jean-François Alcover, Nov 04 2017 *)

c(n) = (4*(n - 2)! + n + 3) / (n^2 - 1);
lista(nn) = forprime(p=2, nn, if (isprime(p+2), print1(c(p+1), ", "));); \\ Michel Marcus, Sep 21 2017

# Python version 2.7
import math
from sympy import *
list = []
n = 3
l = 1   # parameter that indicates the desired length of the list
x = 1
while x <= l:
       y = (4*factorial(n-2))+n+3
       z = n**2 - 1
       if y % z == 0:
              print (y/z)
              list.append(y/z)
       n+=1
       x+=1

a(n) = (4*((p1(n)-1)! + 1) + p1(n))/(p1(n)*(p1(n) + 2)) with p1(n) = A001359(n), for n >= 1. See the name.
From Wilson's theorem (see Hardy and Wright, Theorem 80, p. 68), a(n) = (4*kp1(n) + 1)/(p1(n) + 2) with p1(n) = A000359(n) and kp1(n) = A007619(p1(n)).
a(n) = delta(A014574(n)) with delta(n) = (4*(n-2)!+ n + 3)/(n^2 - 1).
delta(n) ~ ((4*(n-2)^(n - 2)* sqrt(2*Pi*(n - 2))) / (e^(n - 2)*(n^2 - 1)))+((n + 3) / (n^2 - 1)) for large n-values (using Stirling's approximation for n!).

Edited by Wolfdieter Lang, Oct 25 2017

A351423 Expansion of e.g.f. -log(1 - log(1 + log(1 + log(1 + log(1+x))))).

Original entry on oeis.org

1, -3, 16, -124, 1270, -16243, 249776, -4494334, 92716855, -2158505443, 55996266046, -1602132913687, 50124833578256, -1702501170925098, 62391472267252322, -2453892459756494459, 103101294099324376489, -4608723131704380915202

Offset: 1

Seiichi Manyama, Feb 11 2022

sign

Column k=5 of A351420.

Cf. A000359, A351428.

T[n_, 1] := (n - 1)!; T[n_, k_] := T[n, k] = Sum[StirlingS1[n, j] * T[j, k - 1], {j, 1, n}]; a[n_] := T[n, 5]; Array[a, 18] (* Amiram Eldar, Feb 11 2022 *)

my(N=20, x='x+O('x^N)); Vec(serlaplace(-log(1-log(1+log(1+log(1+log(1+x)))))))

T(n, k) = if(k==1, (n-1)!, sum(j=1, n, stirling(n, j, 1)*T(j, k-1)));
a(n) = T(n, 5);

a(n) = T(n,5), T(n,k) = Sum_{j=1..n} Stirling1(n,j) * T(j,k-1), k>1, T(n,1) = (n-1)!.

cp's OEIS Frontend

A003713 Expansion of e.g.f. log(1/(1+log(1-x))).

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A000268 E.g.f.: -log(1+log(1+log(1-x))).

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A000310 Coefficients of iterated exponentials.

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A000406 Coefficients of iterated exponentials.

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A001765 Coefficients of iterated exponentials.

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A039817 Triangle read by rows: matrix 5th power of the Stirling-1 triangle A008275.

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A302358 a(n) = coefficient of x^n in the n-th iteration (n-fold self-composition) of e.g.f. -log(1 - x).

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A292691 a(n) = C(A001359(n)), n >= 1, with C(n) = (4((n-1)! + 1) + n)/(n(n+2)) for n >= 2.